Rinse out the food processor bowl and wipe it dry. Add the softened cream cheese, pumpkin puree, vanilla extract and the remaining 2/3 cup of sugar to the food processor bowl and process for about 30 seconds, or until smooth. Add the eggs and pulse to blend. Add the flour, cinnamon, nutmeg and 1/2 teaspoon of salt and process for about 30 seconds, or until smooth. Carefully pour the pumpkin filling over the gingersnap crust. Set the skillet in the oven and carefully pour in enough hot water to reach halfway up the side of the springform pan. Bake the cheesecake for about 55 minutes, until the filling is barely jiggly in the center. Carefully transfer the skillet to a rack and let the cheesecake cool. Remove the aluminum foil and refrigerate the cheesecake until it is firm, at least 4 hours or overnight.